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On Testing Convexity and Submodularity

  • Michal Parnas
  • Dana Ron
  • Ronitt Rubinfeld
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)

Abstract

Submodular and convex functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well known Monge matrices. A matrix \( V = \left\{ {v_{i,j} } \right\}_{i,j = 0}^{i = n_1 ,j = n_2 } \) is called a Monge matrix if for every 0 ≤i < r ≤n 1 and 0 ≤j < s ≤n 2, we have v i,j + vr,s ≤vi,s + vr,j. If inequality holds in the opposite direction then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix.

In this work we present a testing algorithm for Monge and inverse Monge matrices, whose running time is O((logn1-logrn2/∈), where ∈ is the distance parameter for testing. In addition we have an algorithm that tests whether a function f: [n] → ℝ is convex (concave) with running time of O ((logn)/∈).

Keywords

Property Testing Distribution Matrice Testing Algorithm Combinatorial Auction Submodular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michal Parnas
    • 1
  • Dana Ron
    • 2
  • Ronitt Rubinfeld
    • 3
  1. 1.The Academic College of Tel-Aviv-YaffoIsrael
  2. 2.Department of EE-SystemsTel-Aviv UniversityIsrael
  3. 3.NEC Research InstitutePrinceton

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